"Focus"... can mean several things...
"Focus the lens" so your image, or at least, your point of interest, is sharp or not.
"Expose for the highlights, or the shadows", focus on one. This is a more perceptual meaning. Do you have incredible clouds? You do not want to miss them, expose for the highlights. Do you have a beautiful forest with sunlight passing by? Probably you should focus on the shadows.
What is your point of interest? "Focus on that" and I do not mean center it or focus the lens... I mean compose around it. Take a look at this question: My attention gets repeatedly distracted by the elements needed for the context in this picture. Where am I going wrong?
So In my opinion Focus is a multidimensional concept, and think about it as such. Distance (Z axis) Composition (X and Y axis) Time...
Now some concepts you have a little wrong.
This is simply a mode of the camera to evaluate the relationship between bright and dark areas. This is for exposure.
if everything is sharp
This does not have a direct relation with the previous. Sharp means either in focus or not blurred with for example a vibration of the camera... motion blur.
f/16, everything seems to be sharp to me.
This relates to Depth of Field. Study that concept. There is an additional thing here. Try to use a little wider f stop, like 8 or 11. In some lenses you can experiment some diffraction on smaller apertures.
focus on some specific section (e.g., 6 feet away) what does that mean?
This is confusing... If you HAVE something at 6 feet away, like a branch probably has sense.
Is it ok to switch to live view and zoom to check if things are in focus (not blurred, as many of my photos seems to be a bit blurred)?
Yes and no. It is ok if you are using a tripod, and your situation is not changing. But it is slow.
There is a chance your photos are blurred because the lens quality, smudges, or vibration (motion blur) or simply you missed the focus, because you choose the wrong focus point. Sometimes the lens should be calibrated.